Three-Dimensional Stability Lobe and Maximum Material Removal Rate in End Milling of Thin-walled...

8/2/2019 Three-Dimensional Stability Lobe and Maximum Material Removal Rate in End Milling of Thin-walled Plate

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ORIGINAL ARTICLE

Three-dimensional stability lobe and maximum material

removal rate in end milling of thin-walled plate

Aijun Tang & Zhanqiang Liu

Received: 9 May 2008 /Accepted: 1 August 2008 /Published online: 26 August 2008# Springer-Verlag London Limited 2008

Abstract Chatter phenomenon often occurs during end

milling of thin-walled plate and becomes a commonlimitation to achieve high productivity and part quality.

For the purpose of chatter avoidance, the optimal selection

of the axial and radial depth of cut, which are decisive

primary parameters in the maximum material removal rate,

is required. This paper studies the machining stability in

milling of the thin-walled plate and develops a three-

dimensional lobe diagram of the spindle speed, axial, and

radial depth of cut. Through the three-dimensional lobe, it

is possible to choose the appropriate cutting parameters

according to the dynamic behavior of the chatter system.

Moreover, this paper studies the maximum material

removal rate at the condition of optimal pairs of the axial

and radial depth of cutting.

Keywords Chatter. Three-dimensional stability lobe .

Maximum material removal rate . Thin-walled plate

1 Introduction

With the development of high-speed cutting technology,

many aircraft parts are composed of monolithic components

to form ribs and thin-walled plates. Due to the large area

and low rigidity, the thin-walled plates are always machined

in numerical-control end-milling processes. At the high

removal rate conditions, the milling of thin-walled plate

leads a lot of dynamic and static problems consecutively.

The main dynamic problem is the self-excited vibration

called regenerative chatter. Due to the regenerative chatter,

the cutting thickness is changed with time, which leads tothe dynamic cutting force. Therefore, chatter is one of the

major limitations on high productivity and part quality even

for high-speed and high-precision milling machines.

In workshops, machine tool operators often select

conservative cutting parameters to avoid chatter. Moreover,

in some cases, additional manual operations are required to

clean chatter marks left on the part surface. The studies on

chatter can go back to the 1950s with Tobias [1], Tlusty and

Polacek [2], and Merrit [3], who explained the regenerative

chatter in orthogonal cutting and developed the two-

dimensional stability lobe theory. The two-dimensional

stability lobe theory deals with the stability of solutions

for dynamical cutting systems, which usually stands for the

spindle speed and axial depth. As a function of these two

cutting parameters, the border between a stable cut (i.e.,

chatter-free) and an unstable one (i.e., with chatter) can be

visualized in a chart called stability lobes diagram (SLD).

In the middle of the 1990s, Altintas [4] presented an

analytical form of the stability lobe theory for milling. Both

of these stability lobe theories can help to select the

appropriate cutting parameters of the spindle speed and

axial depth to avoid chatter in machining processes.

In the end milling of thin-walled plates, due to the

change of cutting position, mass and stiffness of cutting

system, and other factors, the two-dimensional stability

lobe diagram sometimes is not comprehensive to describe

the stability of chatter system. Therefore, several researches

on the three-dimensional stability lobe diagram of chatter

system have then been done and achieved some develop-

ments, such as that of Vincent Thevenot et al. who

introduced the dynamical behavior variation of the part

with respect to the tool position in order to determine

optimal cutting conditions during the machining process.

Int J Adv Manuf Technol (2009) 43:3339

DOI 10.1007/s00170-008-1695-y

A. Tang (*) : Z. LiuSchool of Mechanical Engineering, Shandong University,

Jinan, Shandong Province, China

e-mail: [email protected]


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Accordingly, they constructed a three-dimensional stability

lobe diagram of the spindle speed, axial depth, and tool

position with a constant radial depth [5]. U. Bravo and S.

Herranz considered the dynamic parameters variation of the

workpiece along the milling process of the thin-walled plate

as both mass and rigidity were reduced. Sequentially, they

developed a three-dimensional lobe diagram that based on

the relative movement of the chatter systems to cover all theintermediate stages of machining the walls [6, 7]. Tony L.

Schmitz considered the change of system dynamics with

the tool overhang length and developed the three-

dimensional lobe diagram of the spindle speed, axial

depth, and tool overhang length [8].

Material removal rate (MRR) is a critical parameter for

describing the machining process productivity at the

condition of ensuring the quality. MRR in end milling is

proportional to the spindle speed, the axial and radial depth

of cut, the feed per revolution per tooth, and the number of

cutting teeth. Therefore, in order to improve the machining

efficiency, this paper studies the three-dimensional stabilityin milling of the thin-walled plate and develops a three-

dimensional lobe diagram of the spindle speed, and axial

and radial depth of cut. According to the developed three-

dimensional lobe diagram, one can select the appropriate

cutting parameters to avoid chatter at the condition of

ensuring the maximum MRR. The main flow chart of this

research is shown in Fig. 1.

2 Mathematic models of three-dimensional stability lobe

This study is mainly based on the work of Altintas and

Budak [4], and pays more attention on the end milling of

thin-walled plates. Predicting the dynamic behavior for the

chatter system requires a cutting force model, a structural

response model, and a solution procedure.

For simplification, it applies the following assumptions

for the mathematic models of three-dimensional stability of

thin-walled plate:

1. The workpiece is a thin, isotropic, and cantilever plate.

2. The tool is much more rigid than the workpiece.

Therefore, the workpiece can be considered as a

flexible body and can then be modeled as a single

significant mode of vibration

3. The variation of dynamic parameters due to the

material removal is neglected.

4. The end-milling process is up milling, as shown in

Fig. 2.

5. The deflection at different locations of the workpiece is

neglected.

2.1 Mechanical model

According to the assumption 1, the workpiece is modeled

as a thin, isotropic plate, as shown in Fig. 3. I t is

assumed to be clamped at one end, free along the two

parallel sides, and is subjected to the cutting force of the

cutter. It is also assumed that the interactions between the

cutter and the workpiece could be approximately repre-

sented by a cutting region with stiffness k and damping

coefficient c as shown in Fig. 4. The cutter is represented

by a rigid impactor. When the cutter impinges upon thecutting region, the workpiece experiences additional

elastic and damping forces proportional to the cutting

volume.

1. The axial stability limit and spindle speed

In order to analyze easily, the dynamic model in Fig. 4

can be simplified as follows:

1. The chatter system is a linear one.Fig. 1 The main flow chart of this research

Fig. 2 Model of up milling

34 Int J Adv Manuf Technol (2009) 43:3339


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2. The direction of dynamic cutting force is consistent

with that of the steady cutting force, and only the effectof cutting thickness is considered.

The initial dynamic equation can be established accord-

ing to the dynamic model in Fig. 4.

my

t cy

t ky t F t 1

where y(t) is the chatter displacement (mm), F(t) is the

dynamic cutting force (N), m is the modal mass of the chatter

system (Ns2/mm), c is the modal damp of the chatter system

(Ns/mm), and k is the modal stiffness of the chatter system

(Ns/mm).

Equation 1 is treated with Laplace transform. Thetransfer function matrix at the cutterworkpiece contact

zone is defined as follows:

6yy s y s

F s

1

k

1

swn

2 2xs

wn 1

2

where wn ffiffiffi

km

qis the natural pulsation of the chatter

system.

According to the automatic control theory, the time

domain characteristics of the chatter system depend on the

root of the characteristic equation of the transfer function.

When the real part of the root is zero, the system is the

critical state, and the root of characteristic equation can be

expressed as follows:

s iw 3

By substituting Eq. 3 with Eq. 2, the transfer function is

expressed as follows:

6yy iw 1

k

1 r2 i2xr

1 r2 2 2xr 24

where r wwn

, is the chatter pulsation, is the damping

ratio, and x c2mwn

.

On the basis of milling equations of Altinats and Budak,

the transfer function of chatter system is expressed as

follows.

6yy iw 4p

ZapKtayy 1 eiwT 5

where Z is the number of teeth, Kt is the tangential milling

force coefficient (N/mm2), ap is the axial depth of cutting

(mm), and yy is the directional dynamic milling coefficient

in the y direction and can be defined as follows:

ayy 1

2 cos 2f 2Krf Kr sin 2f

fexfst

6

where Kr is the radial milling force coefficient and st, ex

is the start and exit immersion angles of the cutter to andfrom the cutting zone, respectively. In the case of up

milling, the start immersion angle is zero, and the exit

immersion angle is ex. Then, the directional dynamic

milling coefficient can be expressed as follows:

ayy 1

21 cos2fex 2Krfex Kr sin2fex 7

Using the Eqs. 4 and 5, and the Eulers formula, the

axial stability limit is obtained as follows:

ap lim 2pk

ZKtayy

1 r2 2 2xr 2

1 r2

8

Here, the spindle speed ns corresponding to the axial

stability limit can be obtained with the following

expression:

ns 60w

Z 2mp p 2 arctan 2xrr21

9

where m is the whole number of full vibration cycles

between passages of two teeth.Fig. 4 Regenerative chatter model

Fig. 3 Workpiece in end milling

Int J Adv Manuf Technol (2009) 43:3339 35


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2. Radial depth of cut

The start immersion angle of the cutter to the cutting

zone is zero, and the exit immersion angle of the cutter

from the cutting zone is ex. According to Fig. 2, therelationship between the radial depth of cut and exit angle

is defined as follows:

b=R 1 cos fex 10

where b is the radial depth of cutting (mm), and R is the

tool radius (mm).

2.2 Structure response model

Before stability and accuracy predictions can be made, the

dynamic parameters (natural frequency, damping ratio, and

stiffness) of the workpiece for each natural mode must be

developed. The most common method for testing the

dynamic parameters is to impact the structure with an

instrumented hammer and measure the system responsewith an accelerometer. The basic flow chart applied to test

the dynamic parameters is shown in Fig. 5.

Firstly, one needs to exert the exciting force on several

poi nts of the thi n-wal led plate and to mea sur e the

corresponding acceleration. Then, the transfer function be-

tween the impacted points and the measured points can be

obtained by the signal analysis equipment. Lastly, dynamic

parameters of chatter system are derived by curve fitting.

The size of the thin-walled plate used in this paper is

10010010 mm. The material of the part is aluminum

alloy with Youngs modulus of 70 Pa and Poissons ratio of

0.33. Through the developed method, the dynamic param-

eters of chatter system can be obtained. The natural

frequency is 502.5 rad/s, the damping ratio is 0.038, and

the stiffness is 4,500 N/mm.

2.3 Three-dimensional stability lobe

Through the cutting experiments, the tangential milling

force coefficient is 2,400 N/mm2, and the radial milling

force coefficient is 0.9 N/mm2. The cutter is a two-tooth

end-milling cutter. By using Matlab7.0 software, the

stability lobe in three-dimensional of the spindle speed,

axial, and radial depth of cut is represented in Fig. 6.

According to Fig. 6, when the radial depth is half of the

tool radius, the two-dimensional stability lobe diagram of

the axial depth and spindle speed is shown as Fig. 7. Thecurve in Fig. 7 is the boundary between chatter-free

machining operations and unstable process. According to

Fig. 5 Flow chart of testing the dynamic parameters Fig. 7 Two-dimensional stability lobe

Fig. 6 Three-dimensional stability lobe



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Fig. 7, one can select chatter-free combinations of machin-

ing parameters.According to Fig. 6, when the spindle speed is

10,000 rpm, the relationship between the axial and

radial depth under the condition of chatter-free is shown

as Fig. 8. The upper side of the curve in Fig. 8 is

unstable zone, and the down side of that is stable zone.

Figure 8 illustrates that the axial depth at the beginning

gradually decreases with the increment of the radial depth.

When the radial depth increases to a certain extent, the

axial depth begins to increase with the increment of the

radial depth. When the spindle speed is fixed, one can

select the appropriate parameters to avoid chatter accord-

ing to the curve in Fig. 8.

3 Material removal rate

Maximizing productivity is a primary goal in the machine

shop environment. For milling processes, this frequently

translates into maximizing the MRR. Possible mechanisms

for increasing MRR include increasing the spindle speed,

cutting depths, and feed rate. There are practical limits on

the spindle speed and feed rate at which a milling machine

can be operated. Therefore, the increasing MRR can be

achieved through increasing the cutting depth, the axial,

and radial depth of cut.

The mathematic equation for MRR can be expressed as

follows:

MRR ap b ns Z ft 11

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

MRR

(mm*rpm)

Radial depth / Tool radius

105

Fig. 9 Relationship between the radial depth and MRR

0.0 0.1 0.2 0.3 0.4 0.5 0.610

20

30

40

50

60

70

80

90

Exitimm

erseangle

Kr

Fig. 10 Relationship between the ex and Kr

Fig. 11 Relationship between MRR and ex

Fig. 8 Relationship between the axial and radial depth



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where ft is the feed per revolution per tooth. In general, the

effect of ft on stability is small and can be neglected [9].

Hence, MRR is proportional to the multiplication of the

axial and radial depth of cut, and the spindle speed. It is

interesting to find out at which combination of the axial and

radial depth of cut the spindle speed is fixed, where the

maximum value of MRR may be achieved. Therefore, a

normalized value of MRR is used hereafter.

MRRMRR

R ft Z ap

b

R ns 12

If the spindle speed is fixed at 10,000 rpm, at the

condition of chatter-free cutting for the milling of thin-

walled plate with two-tooth end milling, the relationship

between the radial depth and MRR is shown in Fig. 9.

Figure 9 illustrates that MRR is proportional to the radial

depth, and MRR is increased with the higher radial depth.

4 Maximum material removal rate of chatter-free

To obtain the maximum MRR, the radial depth should be

maximal. The acceptable (chatter-free) maximum radial

depth of cut depends on the geometry of the tool, the

direction of cutting, and the mode of milling (up or down

milling).

By substituting Eqs. 7, 8, and 10 with Eq. 12, the MRR

can be expressed as follows:

MRR4pkns

Kt

1 r2 2 2xr 2

1 r2

1 cos fex

1 cos 2fex 2Krfex Kr sin2fex 13

If the geometry and material of the tool, the material of

the workpiece, and the mode of the milling are made sure,

the milling force coefficients Kt and Kr will be certain.

According to the assumption (3) that the variation of

dynamic parameters due to the material removal is

neglected, the modal parameters k, , and r are invariant.

In order to obtain the maximum MRR of the chatter-free,

Eq. 13 is the computed derivatives of the exit immerse

angle ex; the differential equation can be obtained asfollows.

d MRR

dfex

4pkns

Kt

1 r2 2 2xr 2

1 r2

sin fex 1 cos 2fex 2Krfex Kr sin2fex 1 cos fex 2 sin 2fex 2Kr 2Kr cos2fex

1 cos2fex 2Krfex Kr sin2fex 2

14

Ifd MRR

dfex 0, the exit immerse angle ex can be

obtained. From Eq. 14, it is known that the exit immerse

angle is only related with the radial milling force coefficient

Kr. Through Matlab7.0, the exit immerse angle can be

obtained if Kr is made sure. That is to say, for a certain

cutting condition, one can select the appropriate ex to

obtain the maximum MRR of the chatter-free. The

relationship between the exit immerse angle and the radial

milling force coefficient Kr is shown as Fig. 10. Figure 10

illustrates that the exit immerse angle is increased with the

higher radial milling force coefficient. Consequently,

according to the three-dimensional stability lobe, one canobtain that the relationship between the maximum MRR of

the chatter-free and the exit immerse angle ex, which is

shown as Fig. 11. Figure 11 illustrates that the maximum

MRR of the chatter-free is increased with the larger exit

immerse angle ex.

5 Conclusions

Chatter has always been a significant problem in high-

speed machining, which is one of the major limitation on

productivity and part quality. In order to avoid chatter, one

needs to select the appropriate cutting parameters through

the stability lobe. In this paper, a three-dimensional stability

lobe theory has been presented. A three-dimensional, radial

depth of cut has been introduced in the stability lobe

diagram. This method is based on the identification of the

optimal pairs of axial and radial depth of cut. Thus, the

optimal cutting conditions for increased chatter-free MRRcan be obtained through the three-dimensional stability

lobe.

For the purpose of chatter avoidance, the primary

parameters are the axial and radial depth of the cut, which

are decisive in the productivity. Therefore, at the condition



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of stability cutting, one needs to select the optimal pairs of

axial and radial depth to obtain the maximum MRR. If the

geometry and material of the tool, the material of the

workpiece, and the mode of milling are made sure, the axial

and radial depth are both related with the exit immerse angle.

Therefore, in order to obtain the maximum MRR, one needs

to know the appropriate exit immerse angle. This paper

studies the relationship between the maximum MRR and exitimmerse angle, and finds that the maximum MRR of chatter-

free is increased with the larger exit immerse angle.

Acknowledgement The authors are grateful to the National High

Technology Research and Development Program of China for

supporting this work under grant no. 2008AA042405, the National

Natural Science Foundation of China for supporting this work under

grant no. 50675122, and National Science Foundation of Shandong of

China for supporting this work under grant no. Z2007F03.

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4. Altintas Y (2002) Manufacturing automationmetal cutting

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5. Vincent T, Lionel A, Gilles D, Gilles C (2006) Integration ofdynamic behaviour variations in the stability lobes method. Int

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6. Bravo U, Altuzarra O, Lpez de Lacalle LN (2005) Stability limits

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http://dx.doi.org/10.1007/s00170-004-2241-1http://dx.doi.org/10.1007/s00170-004-2241-1http://dx.doi.org/10.1016/j.ijmachtools.2005.03.004http://dx.doi.org/10.1016/j.ijmachtools.2005.03.004http://dx.doi.org/10.1243/095440505X32742http://dx.doi.org/10.1243/095440505X32742http://dx.doi.org/10.1016/j.ijmachtools.2005.03.004http://dx.doi.org/10.1016/j.ijmachtools.2005.03.004http://dx.doi.org/10.1007/s00170-004-2241-1http://dx.doi.org/10.1007/s00170-004-2241-1

Three-Dimensional Stability Lobe and Maximum Material Removal Rate in End Milling of Thin-walled...

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